Abstract
We study the question of uniqueness of weak solution to the fast reaction limit of the Keller and Rubinow model for Liesegang rings as introduced by Hilhorst et al. (2009) [13]. The model is characterized by a discontinuous reaction term which can be seen as an instance of spatially distributed non-ideal relay hysteresis. In general, uniqueness of solutions for such models is conditional on certain transversality conditions. For the model studied here, we give an explicit description of the precipitation boundary which gives rise to two scenarios for non-uniqueness, which we term “spontaneous precipitation” and “entanglement”. Spontaneous precipitation can be easily dismissed by an additional, physically reasonable criterion in the concept of weak solution. The second scenario is one where the precipitation boundaries of two distinct solutions cannot be ordered in any neighborhood of some point on their common precipitation boundary. We show that for a finite, possibly short interval of time, solutions are unique. Beyond this point, unique continuation is subject to a spatial or temporal transversality condition. The temporal transversality condition takes the same form that would be expected for a simple multicomponent semilinear ODE with discontinuous reaction terms.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.